Walsh Spectrum Measurement in Natural, Dyadic, and Sequency Ordering

Muniappan, K.; Kitai, R.

doi:10.1109/temc.1982.304001

Cited by 10 publications

(1 citation statement)

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“…The relationships between the three Walsh function ordering methods (Natural, Strict Sequency and Dyadic) are well known and have been described in references [51], [64], [75] and [84]. The author has observed that by using the previously discovered operations of: bit-reversal for the position of each component in a binary number, Gray code conversion, dyadic addition, and the combination of some of them, one is able to generate not only all the known Walsh orderings from the straight binary code but some new orderings as well.…”

Section: Example Villamentioning

confidence: 99%

Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Falkowski¹

2000

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Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of 2 linear pre-and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment.This dissertation resolves many important issues concerning the efficient application of spectral methods used in the computer-aided design of digital circuits. The main obstacles in these applications were, up to now, memory requirements for computer systems and lack of the possibility of calculating spectra direcdy from Boolean equations.By using the algorithms presented here these obstacles have been overcome. Moreover, the methods presented in this dissertation can be regarded as representatives of a whole family of methods and the approach presented can 1Y: easily adapted to other orthogonal transforms used in digital logic design. Algorithms are shown for Adding, Arithmetic, and Reed-Muller transforms. However, the main focus of this dissertation is on the efficient computer calculation of Rademacher-Walsh spectra of Boolean functions, since this particular ordering of Walsh transforms is most frequendy used in digital logic design.A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON-and DC-cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from non-disjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. By such an approach each spectral coefficient can be calculated separately or all the coefficients can be calculated in parallel. These advantages are absent in the existing methods. The possibility of calculating only some coefficients is very important since there are many spectral methods in digital logic design for which the values of only a few selected coefficients are needed.Most of the current methods used in the spectral domain deal only with completely specified Boolean functions. On the other hand, all of the algorithms introduced here are valid, not only for completely specified Boolean functions, but for functions with don't cares. Don't care minterms are simply represented in the form of disjoint cubes.The links between spectral and classical methods used for designing digital cir- A new algorithm is shown that converts the disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME).Since the known fast algorithm that generates the GR..\1E, based on the factorization of 4 the Reed-Muller transform matrix, always starts from the truth table (mintenns) of a Boole...

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Section: Example Villamentioning

confidence: 99%